nLab 3-groupoid

Context

Higher category theory

higher category theory

Contents

Idea

The notion of 3-groupoid is the next higher generalization in higher category theory of groupoid and 2-groupoid.

Definition

A 3-groupoid is an ∞-groupoid such that all parallel pairs of k-morphism are equivalent for $k \geq 4$: a 3-truncated ∞-groupoid.

Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as equality.

Models

A general 3-groupoid is geometrically modeled by a 4-coskeletal Kan complex. Equivalently – via the homotopy hypothesis-theorem – by a homotopy 3-type.

A small model of this is a 3-hypergroupoid, where all horn-filelrs in dimension $\geq 4$ are unique .

A 3-groupoid is algebraically modeled by a tricategory in which all morphisms are invertible, and by a 3-truncated algebraic Kan complex.

A semistrict algebraic model for 3-groupoids is provided by the notion of Gray-groupoid. These in turn are encoded by 2-crossed modules.

An entirely strict algebraic model for 3-groupoids (which no longer models all homotopy 3-types) is a 3-truncated strict omega-groupoid.

h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | | h-$n$-groupoid | h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid

References

• Simona Paoli, Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids, Journal of Pure and Applied Algebra 211 (2007), 801-820. (arXiv)

• Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv)

Revised on September 10, 2012 20:26:30 by Urs Schreiber (131.174.188.17)